where \(x_1\) , \(x_2\)  are the position operators on \(\ell ^2\big (\mathbb {Z}^2\big )\) , and \(\sigma _1\) , \(\sigma _2\)  are the Pauli matrices.
There are two aspects in the choice of the operator \(\widetilde{D}\)  which are decisive for the construction of the framework of [1]}. First of all \(\widetilde{D}\)  has compact resolvent
as required by the original definition of spectral triple, as given
in [2]}, [3]}.
Secondly, the operator \(\widetilde{D}\)  relates the trace per unit volume \({T}_B\)  with the Dixmier trace \({\operatorname{Tr}}_{{\rm Dix}}\)  [4]}, [2]}, [6]}, [3]}, [8]} via the formula
\({T}_B(A) = \frac{1}{\pi ^2} {\operatorname{Tr}}_{{\rm Dix}}\big (\big |\widetilde{D}_{\varepsilon }\big |^{-2} \widetilde{\rho }(A)\big ) ,\qquad A\in \widetilde{{C}}_B\,,\) 
